Introduction to Complex Numbers

The rules for adding, subtracting, multiplying and dividing complex numbers were worked out by the Italian mathematician Rafael Bombelli (1526–1572), generally credited as the first to develop an algebra of complex numbers (a moon crater is named after him).

Why extend the real numbers at all? Consider two simple equations:

Solving $x^2=c$ in $x$ is the same as finding where the graph of $f(x)=x^2-c$ crosses the $x$-axis.

• For Equation 1, $f(x)=x^2-1$ crosses the $x$-axis at $(-1,0)$ and $(1,0)$: it has the two real solutions $x=-1$ and $x=1$.
• For Equation 2, $f(x)=x^2+1$ never crosses the $x$-axis (its graph sits entirely above it) — Equation 2 has no real solution.

This is because squaring a real number can never produce a negative real number. If Equation 2 is to have a solution at all, we must invent a new kind of number — an imaginary number — as a square root of $-1$. This imaginary unit $\sqrt{-1}$ is denoted $i$, with the defining property $i^2=-1$. Once this single new symbol is admitted, every other power of $i$, and every complex number built from it, follows by ordinary algebra.

Real-life context. Complex numbers describe a phenomenon with two parts varying together at once — e.g. an alternating current has both a magnitude and a phase. Engineers, scientists and vehicle designers who work with electromagnetic signals rely on complex numbers, which also appear in signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography and vibration analysis.

By the end of this chapter you will be able to: perform algebraic operations on complex numbers; plot complex numbers in the Argand plane; find the conjugate and modulus of a complex number; find the polar and Euler form of a complex number; and apply de Moivre's theorem to find the $n$th roots of a complex number.